Relations between Poincaré series for quasi-complete intersection homomorphisms

In this article we study base change of Poincaré series along a quasi-complete intersection homomorphism $\phi \text{\hspace{0.17em}⁣}:Q\to R$, where $Q$ is a local ring with maximal ideal $\mathfrak{m}$. In particular, we give a precise relationship between the Poincaré series ${\mathrm{P}}_M^Q(t)$ of a finitely generated $R$-module $M$ to ${\mathrm{P}}_M^R(t)$ when the kernel of $\phi$ is contained in $\mathfrak{m}{\mathrm{a}\mathrm{n}\mathrm{n}}_Q(M)$. This generalizes a classical result of Shamash for complete intersection homomorphisms. Our proof goes through base change formulas for Poincaré series under the map of dg algebras $Q\to E$, with $E$ the Koszul complex on a minimal set of generators for the kernel of $\phi \mathrm{.}$